D'Alembert's principle
It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert, and Italian-French mathematician Joseph Louis Lagrange.
D'Alembert's principle generalizes the principle of virtual work from static to dynamical systems by introducing forces of inertia which, when added to the applied forces in a system, result in dynamic equilibrium.
[1][2] D'Alembert's principle can be applied in cases of kinematic constraints that depend on velocities.
[clarification needed] Thus, in mathematical notation, d'Alembert's principle is written as follows,
where: Newton's dot notation is used to represent the derivative with respect to time.
The above equation is often called d'Alembert's principle, but it was first written in this variational form by Joseph Louis Lagrange.
[5] D'Alembert's contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish.
The general statement of d'Alembert's principle mentions "the time derivatives of the momenta of the system."
By Newton's second law, the first time derivative of momentum is the force.
However, some applications involve changing masses (for example, chains being rolled up or being unrolled) and in those cases both terms
Consider Newton's law for a system of particles of constant mass,
where Moving the inertial forces to the left gives an expression that can be considered to represent quasi-static equilibrium, but which is really just a small algebraic manipulation of Newton's law:[6]
, done by the total and inertial forces together through an arbitrary virtual displacement,
, of the system leads to a zero identity, since the forces involved sum to zero for each particle.
The original vector equation could be recovered by recognizing that the work expression must hold for arbitrary displacements.
If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces (which is not usually the case, so this derivation works only for special cases), the constraint forces don't do any work,
[7] This leads to the formulation of d'Alembert's principle, which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work:[6]
D'Alembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called "inertial force" and "inertial torque" or moment.
The advantage is that in the equivalent static system one can take moments about any point (not just the center of mass).
Even in the course of Fundamentals of Dynamics and Kinematics of machines, this principle helps in analyzing the forces that act on a link of a mechanism when it is in motion.
In textbooks of engineering dynamics, this is sometimes referred to as d'Alembert's principle.
Some educators caution that attempts to use d'Alembert inertial mechanics lead students to make frequent sign errors.
In such a non-inertial reference frame, a mass that is at rest and has zero acceleration in an inertial reference system, because no forces are acting on it, will still have an acceleration
will seem to act on it: in this situation the inertial force has a minus sign.
[8] D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system.
The result is a set of m equations of motion that define the dynamics of the rigid body system.
of the system as a generalized version of Hamilton's principle for the case of point particles, as follows,
the previous statement of d'Alembert principle is recovered.
[4] For instance, for an adiabatically closed thermodynamic system described by a Lagrangian depending on a single entropy S and with constant masses
Typical applications of the principle include thermo-mechanical systems, membrane transport, and chemical reactions.
